SELECT * FROM question_mgmt as q WHERE id=5148 AND status=1 SELECT id,question_no,question,chapter FROM question_mgmt as q WHERE courseId=9 AND subId=6 AND chapterId=268 and ex_no='1' AND status=1 ORDER BY CAST(question_no AS UNSIGNED) CBSE Free NCERT Solution of 10th mathematics Real Numbers use euclid rsquo s division lemma to show that the

Question:

Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Answer:

Let a be any positive integer and b = 3. Then, by euclid’s algorithm

a = 9q + r, where q ≥ 0 and r = 0, 1, 2                                         [ 0 ≤ r ≤ b ]

For r = 0, x = 3q,   or

For r = 1, x = 3q +1

For r = 2, x = 3q + 2

Now by taking the cube of all the three above terms, we get,

Case (i) :      when r = 0, then,

                     x3 = (3q)3 = 27q3  = 9 (3q3) = 9m; where m = 3q

Case (ii) :     when  r = 1, then,

                     x3 = (3q +1)3 = (3q)3 + 13 +3 × 3q × 1 (3q + 2 ) = 27q3 + 1 +27q2 + 9q

                     Taking 9 as common factor, we get,

                     x3 = 9 (3q3 +3q2 + q) +1

                     Putting  (3q3 + 3q2 + q) = m, we get,

                     x3 = 9m + 1

Case (iii) :     when r = 2, then,

                     x3 = (3q + 2)3 = (3q)3 + 23 + 3 × 3q × 2 (3q + 2) = 27q3 + 54q2 + 36q + 8

                     Taking 9 as common factor , we get 

                     x = 9 (3q + 6q + 4q) + 8

                     Putting (3q + 6q + 4q) = m, we get,

                     x = 9m + 8,

Therefore, it is proved that the cube of any positive integer is of the form 9m, 9m + 1, 9m + 8.


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